Discontinuous Shock Structure in a Reacting Mixture Modelled by Grad 13 Moment Approximation
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Discontinuous Shock Structure in a Reacting Mixture Modelled by Grad 13 Moment Approximation Fiammetta Conforto · Roberto Monaco · Angela Ricciardello
Received: 5 December 2013 / Accepted: 3 April 2014 © Springer Science+Business Media Dordrecht 2014
Abstract A preliminary analysis on the possible occurrence of sub-shocks into a gas mixture is carried out. The mixture, undergoing a reversible bimolecular reaction, is described by macroscopic equations obtained by Grad 13 moment approximation of the reactive Boltzmann equation. Keywords Reacting mixtures · Grad 13 moment approximation · Shock structure
1 Introduction The aim of this paper is to investigate shock structure solutions for a reactive gas mixture described by Grad 13 moment equations, which have been derived from a set of nonlinear Boltzmann equations for a mixture of four gas species undergoing both elastic and chemical interactions [1]. The reactive process consists in a reversible bimolecular reaction [2] of the form 3 + 4 1 + 2. Discontinuous travelling wave solutions for a hyperbolic dissipative system have been studied in [3]. The authors have considered such a problem for a monatomic gas described by a 13 moment closure derived by extended thermodynamics, with the aim of connecting,
This work is sponsored by GNFM-INdAM, University of Messina and Torino Politecnico.
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F. Conforto ( ) Dept. of Mathematics and Computer Science, University of Messina, V.le F. Stagno d’Alcontres 31, 98166 Messina, Italy e-mail: [email protected] R. Monaco DIST, Politecnico di Torino, V.le Mattioli 39, 10125 Torino, Italy e-mail: [email protected] A. Ricciardello Faculty of Engineering and Architecture, University KORE of Enna, Cittadella Universitaria, Via delle Olimpiadi, 94100 Enna, Italy e-mail: [email protected]
F. Conforto et al.
through a discontinuous travelling wave, two equilibrium states on different sides of the barrier representing the locus of singular points of the system. In this paper we present a generalization of such a problem to the case of a reacting mixture of four gases. In the case of a mixture the barrier is decomposed in four different sub-manifolds of singularities, one for each species. Consequently, a discontinuous travelling wave may exhibit one or more sub-shocks depending on the number of sub-manifolds lying between the two equilibria. In Sect. 2 we present Grad’s hydrodynamic equations and derive the conservation equations useful to deduce the relations between the two equilibria. The formulation of the problem, its preliminary analysis and some numerical inspections are then included in Sect. 3.
2 Grad’s 13-Moment Equations In 1-D the distribution functions fi , i = 1, . . . , 4, of the gas species are supposed to have spherical symmetry around z-axis in the phase space. The closure of the reactive Boltzmann equations is achieved by using 13 moment Grad distribution functions [1] σi m i 2 4 qi mi mi 2 5 ci + ci ci − fi (v) = fMi (v) 1 + ni kB Ti 2kB Ti 5 ni kB Ti 2kB Ti 2kB Ti 2 where
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